Approximating Free-form Geometry with Height Fields for
Manufacturing
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Citation
Herholz, Philipp, Wojciech Matusik, and Marc Alexa.
“Approximating Free-Form Geometry with Height Fields for
Manufacturing.” Computer Graphics Forum 34, no. 2 (May 2015):
239–251.
As Published
http://dx.doi.org/10.1111/cgf.12556
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Wiley Blackwell
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Author's final manuscript
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Wed May 25 22:45:06 EDT 2016
Citable Link
http://hdl.handle.net/1721.1/100915
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EUROGRAPHICS 2015 / O. Sorkine-Hornung and M. Wimmer
(Guest Editors)
Volume 34 (2015), Number 2
Approximating Free-form Geometry with Height Fields
for Manufacturing
Philipp Herholz1
1 Technische
Wojciech Matusik2
Marc Alexa1
Universität Berlin 2 MIT CSAIL
Figure 1: To fabricate a mesh, possibly acquired by laser scanning, in a molding process, we automatically decompose and
deform the shape (center left). A reusable multi-piece mold for the object can be produced by CNC milling (center right).
Physical copies can be fabricated by mold casting in different materials like resin or plaster (right).
Abstract
We consider the problem of manufacturing free-form geometry with classical manufacturing techniques, such as
mold casting or 3-axis milling. We determine a set of constraints that are necessary for manufacturability and
then decompose and, if necessary, deform the shape to satisfy the constraints per segment. We show that many
objects can be generated from a small number of (mold-)pieces if slight deformations are acceptable. We provide
examples of actual molds and the resulting manufactured objects.
1. Introduction
The recent development in 3d printing has liberated digital manufacturing: today a large variety of shapes can be
printed and design is in many cases no longer constrained
by the manufacturing process. Nonetheless, 3d printing has
still a number of problems, some of them inherent to the
technology. For example the size of the production volume
is severely limited for most technologies, production time
is high and so is the price per unit. Another issue is the limited choice of materials, surface finishes, and structural properties. Classical manufacturing techniques such as casting,
stamping, and 3-axis milling are in many aspects complementary. These technologies are applicable for a large variety of materials. Since they are targeted at mass production,
cost per item and production times are generally low once a
mold or die has been manufactured. On the other hand, these
c 2015 The Author(s)
c 2015 The Eurographics Association and John
Computer Graphics Forum Wiley & Sons Ltd. Published by John Wiley & Sons Ltd.
technologies are severely limited in terms of shapes that can
be produced. For most classical fabrication techniques the
shape has to be a height field. In 3-axis milling, for example,
the drill bit has to be able to reach every point on the surface.
The same is true for stamping, a technique commonly used
in the automotive industry to press metal sheets into a required shape. Another example is mold casting, where two
or more molds form a cavity that is filled with liquid material. After the material has hardened, the mold pieces have
to be removed by a linear translation, hence, again yielding a
height field constraint for individual mold pieces. There are
two common approaches to respect the constraint on shapes
to be a height field. Either shapes have to be designed in such
a way that they constitute height fields, or they have to be divided into parts that fulfill the constraint individually. So far,
specialized engineers are needed to segment a given design
Herholz et al. / Approximating Free-form Geometry with Height Fields for Manufacturing
into parts that can be fabricated by classical manufacturing
techniques.
Our objective is to automate this process. In contrast to industrial products, we target the fabrication of free-form geometry, possibly acquired by laser scanning. Industrial parts
are typically designed with manufacturability in mind and
mold-makers and designers work in a loop, iteratively improving the design. This case is covered by a large body
of literature in CAD/CAM [PG04, KBM06, CR09]. Our situation, on the other hand, is more involved, since we oftentimes deal with organic shapes having very intricate features and concavities. However, we are not concerned with
functional parts, like gears, that have to fit perfectly into an
assembly. This provides us with some additional freedom:
we can deform the shape slightly in order to meet the constraints. This deformation often enables a significant reduction in the number of parts that are required while typically
only leading to small visual distortion.
We also explain how global constraints on mesh segmentation for fabrication, like total seam length and constraints
on the maximal build volume, can naturally be incorporated
into our system. Moreover, we provide an interactive user interface, enabling users to manually specify local constraints,
such as the amount of allowed deformation.
Contributions of this work include:
• A method to decompose a shape into surface patches that
largely satisfy the height field constraint. Constraint violation can be controlled and even completely prevented.
• An adapted version of as-rigid-as-possible mesh deformation to remove local constraint violations.
To demonstrate the practicability of our approach, we fabricated physical mold pieces and used them to cast several
input meshes. In addition, we fabricated examples using 3axis CNC milling.
2. Related work
2.1. Mold design
Mold design is a prominent topic in the engineering literature. Naturally, these works are targeted at shapes already
designed for fabrication. Therefore, the requirements and assumptions on the input are quite restrictive. Ravi and Srinivasan [RS90] give a list of decision criteria for mold design,
like undercuts, flatness, and stability. Based on these criteria Chakraborty and Reddy [CR09] devise an algorithm for
the construction of two-piece permanent molds with side
cores. Pryadarshi and Gupta [PG04] try to find a decomposition into multi-piece molds. They consider the problem as a set cover problem. Concurrent with our analysis of set cover techniques, the method generates a large
number of mold pieces for free-form surfaces (cf. Figure
4). Closely related to our approach is the work of Li et
al. [LML09]. Given a casting direction, they determine an
optimal parting line. Undercuts are removed by a mesh deformation technique. They consider only two-piece molds,
however, and manufacturability is not guaranteed. Khadekar
et al. [KBM06] employ the GPU to display undercuts by
rendering the model from potential casting directions. This
procedure is extremely fast and allows for a quasi exhaustive
search of the space of possible directions. However, we have
found this method to be unreliable for general meshes with
small triangles since these may be discarded during rendering. Very recently He et al. [HLZCO14] explored the decomposition of shapes into pyramidal pieces. A shape is pyramidal if it has a flat base and the rest of the shape is a height
function over that base. They consider molding and reducing
the amount of support material in fused deposition printing
as applications. Our approach is more general since we do
not require the parts to have a flat base.
2.2. Mesh processing for fabrication
In recent years, different works in computer graphics addressed mesh processing for fabrication. Luo et
al. [LBRM12] focus on the build volume constraint of 3d
printers and try to find a segmentation of a model into a small
number of parts such that the assembled object is structurally sound and hides seams. Our approach also allows to
consider a build volume constraint and can hence also help
to overcome the problem addressed in this work. Julius et
al. [JKS05] segment a shape into almost developable patches
to enable the fabrication from sheets of materials, for example, for sewing stuffed toys from sheets of fabric. Similarly, Chen et al. [CSaLM13] simplify a model until it consists of only a few polygons that can be fabricated with a
laser cutter. Both works aim at 2-axis production devices
and hence require more severe changes to the model than
necessary with our approach. McCrae et al. [MSM11] and
Hildebrand et al. [HBA12] decompose a shape into orthogonal planar pieces that resemble the input model. However,
models generated with this approach are only abstractions
of the original input while our techniques yields replicas,
possibly up to a small error due to deformations. Vanek et
al. [VGB∗ 14] segment a model into pieces to save production time and support material during 3d printing by packing pieces tightly into the production volume. Our technique
is complementary and saves support material by avoiding
the overhangs that make support necessary in the first place.
Similar to our work, mesh deformation has also been used
in other works to improve fabrication properties of 3d objects. For example, Stava et al. [SVB∗ 12] used deformation
to ensure the structural stability of fabricated objects. Prevost et al. [PWLSH13] employ deformation for adjusting the
shapes’ center of mass such that it can stand upright without
falling over.
c 2015 The Author(s)
c 2015 The Eurographics Association and John Wiley & Sons Ltd.
Computer Graphics Forum Herholz et al. / Approximating Free-form Geometry with Height Fields for Manufacturing
3. Problem statement
We assume the input geometry is given as a triangle mesh
M = (V, F) with vertex coordinates V = {vi } and triangles
specified by their vertex indices F = {(i, j, k)}. The mesh
is assumed to represent a discrete, oriented manifold with
or without boundary. Our goal is to create a physical copy
of the shape enclosed by M using classical manufacturing
techniques such as 3-axis milling, casting, and stamping. All
of these techniques constrain the shape to be a height field
with respect to a specific fabrication direction d: stamping
can only produce thin height field sheets; 3-axis milling can
only be used to fabricate a height field; casting techniques requires the portion of the shape associated with an individual
mold piece to be a height field. In order to use these techniques we segment the model into disjoint connected height
field components Ci :
Ci ⊆ F ,
[
Ci = F,
Ci ∩C j = ∅.
Figure 2: Global overlaps will occur if the projected boundary polygon has self-intersections.
(1)
i
The sub-meshes Ci can then be manufactured individually
and assembled to form the input shape. In the case of multipiece molding, the segments can be used to produce mold
parts which in turn can be used to manufacture the input
shape in one cast. Let nt be the outward pointing surface
normal of triangle t ∈ Ci and di a potential fabrication direction. A necessary condition for Ci to be a height field with
respect to di is
dT
i nt
≥ 0,
∀t ∈ Ci .
(C1)
This is, however, not sufficient for Ci to be a height field
since global overlaps can still be present (c.f. Figure 2). Lipman [Lip13] has shown that if all triangles satisfy (C1) then
it is sufficient that the boundary polygon(s) of Ci , projected
along di , are simple, that is, there are no self intersections.
Let Pi be the projection along di , i.e., Pi = I − di dT
i . Then
we can verify the boundary condition by
Pi vj0 , Pi vj1 ∩ Pi vk0 , Pi vk1 = ∅.
(C2)
for all edges ( j0 , j1 ), (k0 , k1 ) ∈ ∂Ci of the boundary ∂Ci . See
Figure 2 for an example of a surface patch that obeys (C1)
but violates (C2).
In the following we consider several additional constraints
that might be necessary depending on the manufacturing device or application scenario.
3.1. Shell constraint
In multi-piece molding each mold piece has to be removed
from the cast after the material has cured. If face normals of
a segment use the full half-space of valid normals with respect to the fabrication direction, a linear translation along
this direction is the only way to remove the mold piece.
The above constraints (C1) and (C2), however, cannot guarantee that it is possible to remove the mold pieces since
interlocking can occur. Consider for example two adjacent
c 2015 The Author(s)
c 2015 The Eurographics Association and John Wiley & Sons Ltd.
Computer Graphics Forum Figure 3: Illustration of the shell constraint. The two segments can not be removed by a translation along their fabrication directions d(Li ) and d(L j ).
pieces of a 3d-jigsaw puzzle. Even though the curved surface of each piece might fulfill (C1) and (C2) individually,
the pieces can not be disassembled
by a linear motion along their fabrication direction. Figure 3 illustrates
such a situation. To detect interlocking of adjacent mold pieces we consider tangents of the surface orthogonal to seam edges. As shown in the
Figure to the right, the tangent vector
t j1 , j2 , j0 at edge ( j1 , j2 ) of the triangle
( j0 , j1 , j2 ) is given by
t j1 , j2 , j0 =
n j0 , j1 , j2 × (v j1 − v j2 )
.
||n j0 , j1 , j2 × (v j1 − v j2 )||
(2)
Interlocking can be prevented by enforcing
dT
i t j1 , j2 , j0 ≥ 0,
( j0 , j1 , j2 ) ∈ Ci
(C3)
for all seam edges ( j1 , j2 ). To see this, consider extruding
a surface component into a shell piece along the normal at
each vertex. Boundary edges are extruded parallel along the
face normal, forming planar quadrangular polygons. These
polygons will form the contact surface with neighboring
shell pieces. The normal of such a boundary polygon extruded from edge ( j1 , j2 ) is t j1 , j2 , j0 (Figure 3, right). To assemble the shape by moving the shell piece of component Ci
towards another one in direction di the contact surface has
to be completely visible from direction di . This is equivalent
to (C3).
Herholz et al. / Approximating Free-form Geometry with Height Fields for Manufacturing
3.2. Volume constraint
Most manufacturing devices have a cuboid build volume
with side lengths (dxmax , dymax , dzmax ). We hence require
that for every component Ci there is a bounding box with dimensions dx × dy × dy contained in dxmax × dymax × dzmax :
dx ≤ dxmax ,
dy ≤ dymax ,
dz ≤ dzmax .
(C4)
3.3. Soft constraints
The fundamental principle of our approach is to combine
segmentation and deformation to find a compromise between the number of parts and the necessary deformation
of the geometry. One can consider this as additional constraints, i.e. keep the number of components Ci small and
minimize d(V,V 0 ) under some appropriate metric d. Note,
that these constraints are conflicting; allowing for more deformation will reduce the number of parts and vice versa.
We employ a user set parameter to balance these objectives.
3.4. Other possible constraints
There are other constraints specific to certain manufacturing
devices that do not model here. For examples, we currently
do not consider the finite tool size of milling machines. Similarly, we currently do not take global assembly constraints
for multi-piece molds into account. This means that while
we do enforce that faces fi are not occluded in their own
segment Ck 3 fi , we do not require faces to be globally rayaccessible. This means that a ray from fi into direction dk
might hit another face of the model. Consequently, mold
pieces might not be removable. We feel it is often better to
consider such problems after processing the shape and find
solutions with the user in the loop in an interactive system.
Reasons and consequences of this design decision will be
discussed further in Section 9.1.
4. Method
In this section we present our approach for solving the fabrication problem formulated in the last section. First, we uniformly sample a large number of potential fabrication directions di on the Gauss sphere. For each direction di we mark
all triangles that satisfy constraint (C1) w.r.t. this direction
and add all connected components of these triangles to the
set {L j } . Since there can be more than one component per
direction di , we use the notation di = d(L j ) to denote the
direction associated with connected component L j .
The main idea of our segmentation procedure is to cover
the shape with a small number of components from {L j },
such that all constraints are satisfied. It might appear that a
solution to our segmentation problem can be obtained using
approximations to the set cover problem. Set cover approximations would select few components whose union cover
Figure 4: A solution to the segmentation problem provided
by an approximation to set cover generates too many individual components to be useful.
the surface. However, they lack any information on how
these components might overlap and how they might become disconnected as a result (see Figure 4). This is impractical since each connected component of the set cover
would have to be manufactured individually. Moreover, it is
not straightforward to incorporate (C2) and (C3) into these
algorithms. Hence, we rather employ a graph cut based approach which is described in the next section. This allows us
to model the problem to obey all necessary constraints while
still permitting fast approximations.
4.1. Constraint relaxation
For many shapes, especially ones obtained by scanning real
world objects, trying to cover the surface with the components {L j } does not lead to a manageable number of parts,
say two to seven, even if we ignore all other constraints.
Consider for example the hair structures on the model ’lion
vase’ (Figure 15). No matter how many directions we sample, every face set will inevitably contain many holes. We
observe that a small amount of deformation is in many cases
enough to significantly reduce the number of required parts.
One possible approach would be to deform the shape in a
preprocessing step. However, without knowledge of a specific segmentation it is hard to tell how the mesh should be
deformed in order to reduce the number of components. Our
strategy for minimizing the resulting deformation is to relax the constraints in the segmentation phase (Section 5) and
correct for possible constraint violations in a post-processing
step (Section 6).
Crucial for the success of our approach is how the constraints are relaxed. We process each face set L j individually and record connected components of the complement
F \ L j . For each component in this complement we compute
the (unsigned) projected area w.r.t. d(L j ), which provides
us with an a priori estimate for the necessary deformation.
We add a component of F \ L j to L j only if its projected
area is below a threshold, which we denote γ. This parameter controls the deformation necessary to make the shape
manufacturable. Quite intuitively, allowing more deformac 2015 The Author(s)
c 2015 The Eurographics Association and John Wiley & Sons Ltd.
Computer Graphics Forum Herholz et al. / Approximating Free-form Geometry with Height Fields for Manufacturing
tion usually leads to fewer parts, so γ can be used to balance
the number of parts and how close the manufactured shape
matches the original geometry. We demonstrate this effect in
Section 8.4.
components associated with any of the sampled directions.
Every triangle f ∈ F is contained in many of these sets. The
set L( f ) if formed by indizes of face sets that contain f :
4.2. Enforcing the volume constraint
Our objective is to choose for every triangle f one of the labels from L( f ). Every labeling li : fi → {1, 2, · · · } such that
li ∈ L( fi ), induces a valid segmentation with respect to (C1).
Note that enforcing (C2) requires checking pairs of seam
edges which is a global problem, complicating the search for
a valid labeling (see Section 5.2). Therefore, we restrict the
space of valid labelings even further. For each face set L j we
record all faces that are either flipped or occluded by other
faces from L j in direction d(L j ). This can be achieved efficiently by projecting L j onto the plane orthogonal to d(L j )
and recording triangle intersections in 2d. If a face fi ∈ L j is
occluded or flipped, the labeling li = j is only allowed if all
adjacent faces to fi also get the label j. In other words, none
of the three triangle edges is allowed to become a seam edge
of the segmentation. This is a rather conservative approach
since the occluder is part of L j but might not be part of the
segment Ck 3 fi . Therefore the edges of fi might not be occluded from another seam edge in Ck at all. Checking for
(C3) is a simple local task. Suppose ( j0 , j1 ) is a seam edge
with two adjacent faces ( j0 , j1 , j2 ) and ( j1 , j0 , j3 ) carying
labels k and l respectively. This labeling is only valid if
Since a subset of a face set Lk will potentially form a surface segment Ci , the spatial extent of Lk represents an upper
bound on the extent of Ci . To enforce a cuboid build volume constraint we trim face sets exceeding the build volume. To maximize the area of the trimmed face set we first
construct a bounding box that is aligned with the principal
directions and the fabrication direction of the face set. The
bounding box is voxelized and the enclosed surface area for
each voxel is determined. This information is used to find the
axis aligned build volume cuboid maximizing enclosed surface area. This problem can be solved efficiently by dynamic
programming.
This approach is of course conservative since the segments can be trimmed in many different ways to obey the
volume constraint. Another approach would be to enforce
the build volume constraint in the optimization phase. Since
the constraint is global in nature, this would require a more
complex optimization procedure (Section 5.2).
L( f ) = { j | f ∈ L j }.
d(Ll )T t j0 , j1 , j2 ≥ 0
4.3. Adaptive direction sampling
Sampling candidate directions di uniformly works well
for many shapes. However, preferable directions might be
missed. A preferable direction can be characterized as a
direction with a connected component L j of large surface
area. Since sampling and processing a very large number
of directions is computationally demanding, we propose an
adaptive sampling strategy. Starting with a coarse sampling,
we compute connected components of face sets for each
direction. Based on the maximal surface area per direction we refine the sampling
adaptively with respect to surface area. Figure 5 illustrates
maximal surface area per direction color coded on the
sphere. Sampling positions
are marked as black points,
Figure 5: Area per sampled local maxima w.r.t. surface
area as red points.
direction.
5. Segmentation
We consider mesh segmentation as a discrete labeling problem. Connected components of faces carrying the same label
will form segments Ci which can be manufactured individually. Our input are the face sets {L j } formed by connected
c 2015 The Author(s)
c 2015 The Eurographics Association and John Wiley & Sons Ltd.
Computer Graphics Forum and
d(Lk )T t j1 , j0 , j3 ≥ 0.
(3)
(4)
5.1. Energy minimization
We formulate the problem of finding a valid segmentation as
an energy minimization problem defined over the edges and
nodes of the face graph GF . The faces constitute the nodes
of this graph and graph edges represent edges shared by two
faces. Our energy is designed such that a multi-label graph
cut algorithm [BVZ01] can efficiently find an approximate
minimizer. For any labeling (li )i∈F the energy is given by
E=
∑ Ei (li ) +
i∈F
∑
Ei, j (li , l j ) + λ |{li , i ∈ F}| (5)
(i, j)∈F ×F
where, by abuse of notation, F is used as an index set over
all faces of M. The unary term Ei (li ) represents the cost
associated with setting the label for face fi to li . We use this
term to enforce that the label for fi is valid:
(
0
if li ∈ L( fi )
Ei (li ) =
(6)
∞ otherwise.
Minimizing Ei alone will yield a valid labeling with respect
to (C1). However, the boundary might not have an injective
projection (C2) and seam edges might violate (C3). Moreover, result in the use of many different labels, leading to
a large number of segments Ci . The pairwise term Ei, j in
Eq. (5) helps to alleviate both problems. We use the follow-
Herholz et al. / Approximating Free-form Geometry with Height Fields for Manufacturing
ing definition:
Ei, j (li , l j ) =


0
if li = l j ,





∞
if fi or f j violates (C1),

∞
if fi or f j is occluded in Lli or Ll j ,



∞
if the edge fi ∩ f j violates (C3),




| fi ∩ f j | otherwise.
has to hold:
Ei, j (α, α) + Ei, j (β, γ) ≤ Ei, j (α, γ) + Ei, j (β, α).
(7)
In the context of Markov random fields, Ei, j is frequently
called smoothness term, since it encourages a consistent
choice of labels for neighboring faces. The cost of an inconsistent labeling for two faces connected by an edge is the
length of that edge. Edges violating (C2) or (C3) with respect to a specific labeling of their adjacent faces lead to an
infinite energy, therby enforcing the constraints. The value
of Ei, j for non-adjacent faces fi and f j is always zero, the
energy is therefore local.
The sum of Ei, j in Eq. (5), running over all edges in the
mesh, measures the total seam length, if it has a finite value.
Note that minimizing the total seam length also encourages
the use of a small number of labels. By setting λ to a nonzero
value it is, however, possible to explicitly punish the use of
many different labels.
5.2. The α-expansion algorithm
We give a brief overview of the optimization algorithm used
in our implementation. The α-expansion algorithm [BVZ01]
finds an approximate minimizer of the energy E by iteratively computing graph cuts. The graph used by the algorithm is an augmented version of the input graph GF , to
unify the treatment of unary and binary energies. Starting
from an initial labeling the algorithm proceeds by focusing
at one label α at a time. A graph cut answers for every node
the question whether to retain its label or to switch to α. This
process, called α-expansion, is repeated by cycling through
all labels until convergence.
(8)
Furthermore, the energy has to be positive and symmetric
with respect to its arguments.
The condition in Eq. 8 prevents us from incorporating
global constraints, like limited build volume, directly into
the energy minimization. If we want, for example, to prevent two non-adjacent faces to end up in the same segment
because they occlude each other, we could try to add a new
edge between these faces in the graph and set Ei, j (li , l j ) = ∞
for li = l j and 0 otherwise. Unfortunately, this would violate
the condition stated in Eq. 8. This is the reason why we chose
to forbid seam edges that are occluded by a face from the label face set, even though the occluder might not be part of
the actual surface patch. For the same reason the maximum
build volume constraint is enforced by trimming label face
sets L j .
5.3. User interface
We provide a simple interactive user interface to help the
user guide the segmentation process, if desired. The user
might want to prevent seams to cut through specific regions
of the surface or considers deformation unacceptable for
salient features. Our interface enables the user to mark faces
with a brush tool as ’no seam region’ or ’no deformation
region’. These additional constraints are enforced by modifying the unary and binary energy terms. For faces Fi marked
as ’no seam region’ the term Ei, j is set to ∞ for all neighboring faces Fj . Faces that are not supposed to be deformed
are only allowed to be given a label if they obey the hard
constraint (C1) with respect to the label direction. The user
defined constraints have to be consistent; disallowing seams
for the whole mesh will not yield a valid result with finite
energy E. We therefore check if the constraints are feasible
and give a warning to the user if this is not the case. After
The quality of the result is influenced by two factors: the
initial labeling and the order in which labels are processed.
While the solution is largely independent of the initial labeling, for which we use a random assignment, the order of the
labels has a great influence on the approximation quality of
the solution. After assessing several strategies for generating
’good’ label orders to speed up convergence, we found that
running the algorithm several times for completely random
label orders gives consistently the best results.
Minimizing labeling problems with general unary and binary energy terms is NP-hard. To find approximate minimizers of these energies, the α-expansion algorithm relies
on graph cuts. This approach comes with a limitation: the
binary energy term Ei, j term has to be a semi-metric. Therefore, for three labels α, β, γ the following triangle inequality
Figure 6: Our algorithm optionally supports user interaction. The user might not be satisfied with the initial segmentation (left). To prevent seams from cutting through the face,
the user marks the region as ’no seam region’ (center). Running more α-Expansion iterations with the modified energy
yields a result that satisfies the user constraints (right).
c 2015 The Author(s)
c 2015 The Eurographics Association and John Wiley & Sons Ltd.
Computer Graphics Forum Herholz et al. / Approximating Free-form Geometry with Height Fields for Manufacturing
the user made changes to the constraints either a new run of
the α-expansion algorithm is performed or additional α expansion iterations, based on the current labeling. We refer to
the accompanying video for a demonstration.
6.2. Shape energy
Our deformation energy follows the ARAP approach, i.e. we
wish to minimize
E(V 0 , R0 ) =
∑ ∑
wtjk ||v0j − v0k − Rt0 (v j − vk )||2
(13)
t∈Ci j,k∈t
6. Deformation
After the mesh has been segmented, some faces might violate constraint (C1), while all component boundaries satisfy
(C2) and (C3) (if desired). Figure 7(a) shows faces violating
(C1), color coded for a particular component and direction.
Our goal is to deform the mesh as little as possible in order
to satisfy the constraint (C1) without degrading the overall
appearance of the mesh or violate the boundary constraints
(C2) and (C3).
A simple way to satisfy (C2) and (C3) is to fix the boundaries of each component. This also lets us treat each component independently. In the following we consider a single
component Ci .
where wtjk is the cotan weight for edge ( j, k) of triangle t.
In contrast to [SA07] we use per face transformations rather
than per vertex ones. Not considering any constraints, the
main feature of this energy is that it can be easily minimized
by alternating between minimizing w.r.t. V 0 , which requires
solving a linear system, and minimizing w.r.t. the rotations
{Rt0 }, whose solution is Rt0 = Rt , as defined above.
In the usual setup, a subset of vertex positions is constrained, so that constraints are taken care of in the global
step. The main difference in our setup is that we need to
minimize the energy such that constraints on the rotations Rt
in Eq. (12) are satisfied. Note that if E is minimized, the rotations Rt0 have converged against Rt so that satisfying Eq. (12)
really means V 0 satisfies (C1).
6.1. Energy formulation
6.3. Lagrangian setup
We face the task of fixing the geometry to comply with (C1)
as an energy minimization problem. To this end we formulate a constrained version of the well known as-rigid-aspossible (ARAP) surface deformation energy [SA07].
Let the original geometry of component Ci be represented
by the vertex set V and the deformed geometry by V 0 . We
can describe the transformation of triangle t = ( j0 , j1 , j2 ) as
follows: Let Et be the matrix of edge vectors of t, i.e.,
Et = v j1 − v j0 , v j2 − v j1 , v j0 − v j2 ,
(9)
and Et = Ut Σt WtT the SVD of this matrix. Furthermore, let
T
Et0 = Ut0 Σt0Wt0 be analogously defined for the deformed geometry. Then
We suggest to optimize an approximation of the Lagrange
dual form [BV04] of the constrained optimization problem:
!
max inf
E(V 0 , R0 ) −
0
0
λ V ,R
∑ λt dTi Rt0 nt
.
(14)
t∈Ci
This is only an approximation because the constraint is formulated in terms of the variable Rt0 rather than in the rigid
part Rt of the actual transformation At , which would depend
on V 0 . We discuss the consequences of this approximation
later.
(10)
Computing the infimum w.r.t. V 0 is independent of λ and
yields the same linear system as when E(V 0 , R0 ) is minimized. Optimizing for the rotations R0 is still a local problem, i.e., we can optimize each pair Rt0 , λt independently.
is a linear transformation mapping the edges of t from the
original to the deformed state. More specifically, it is the
unique transformation with the property that the normal nt
maps to zero. Moreover
The KKT conditions for a local optimum require that
0
λt ≥ 0 and λt dT
i Rt nt = 0. The first condition leads to
T 0
di Rt nt ≥ 0, which together with the second condition yields
exactly two cases:
At = Ut0 Σt0Wt0TWt Σt+UtT
Rt = Ut0Wt0TWt UtT
(11)
is the closest orthogonal transformation to this linear map
(with respect to the Frobenius norm). Note that the SVD of
the undeformed geometry needs to be computed only once
when considering different deformed geometries.
The constraint (C1) for the deformed geometry V 0 can
now be described in terms of the rigid part of At , namely
Rt , as
T 0
dT
i Rt nt = di nt ≥ 0,
t ∈ Ci
(12)
since At does not have any effect on nt .
c 2015 The Author(s)
c 2015 The Eurographics Association and John Wiley & Sons Ltd.
Computer Graphics Forum 0
1. dT
i Rt nt > 0: This means λt = 0, and the minimization is
identical to the standard ARAP case, i.e. the optimum is
found for Rt0 = Rt .
0
0
2. dT
i Rt nt = 0: This constrains Rt so that it rotates nt into
the plane orthogonal to di . Among these rotations the optimal one minimizes E(V 0 , R0 ), which means it is the rotation from this set closest (in the Frobenius norm) to At .
This is equivalent to
min
kRt0 − Rt kF ,
0
Rt
0
dT
i Rt nt = 0.
(15)
Note that if Rt rotates nt into nt0 , we wish to further rotate nt0 so that it is orthogonal to di . The shortest path
Herholz et al. / Approximating Free-form Geometry with Height Fields for Manufacturing
(a)
(b)
(c)
Figure 7: Constraint violations have to be removed after segmenting the shape (a). Deforming the shape using constraint
rotations yields a mesh that is almost a height field (b). Remaining error can be eliminated by remeshing (c).
for this rotation results from rotating around di × nt0 . The
0
necessary angle of rotation is cos−1 (1 − dT
i nt ). Denote
the rotation matrix with this axis and angle Rt∗ then the
optimal transformation is Rt0 = Rt∗ Rt .
In order to compute the optimal transformation, one first
0
computes Rt , checks if dT
i Rt nt > 0 and than sets Rt = Rt
if true and Rt0 = Rt∗ Rt else.
6.4. Optimization
6.5. A-priori estimation of the deformation
Recall that we need to estimate whether a region of triangles
in a component can be fixed easily. We use the Lagrangian to
compute an estimate of the necessary deformation, without
actually performing it.
At initialization, the triangles contributing to the Lagrangian are exactly the ones violating the constraint. For
the initial step, we set the transformation for these triangles
to Rt0 = Rt∗ (note that Rt = I at initialization). We can evaluate
E(V, R0 ) =
0
0
Repeating the minimization steps for V and R leads to a
steady state. For most faces we find Rt0 = Rt meaning that the
constraint (C1) is necessarily satisfied. For some faces we
have Rt0 = Rt∗ Rt in the converged case meaning it is unclear
if, indeed, dT
i Rt nt ≥ 0.
Note that the meshes we subject to this process require
rather small deformations by design. This means the least
squares fit of triangles to Rt0 leads to very small deviations
from the constraint. In fact, in most examples in the paper
we have taken the mesh resulting from this optimization and
directly used it for manufacturing (e.g. computing the tool
path for the CNC mill).
In case the violation seems larger than acceptable (this
will depend on the application), we suggest the following
remeshing procedure: project the surface patch on the plane
orthogonal to di and delete the violating triangles. The resulting holes can be triangulate in the plane (we use Triangle [She01] for this task). Finally, we assign height values
along di for the newly added vertices based on interpolation.
This necessarily fixes all possibly remaining problems. It is
of course also possible to use the remeshing method directly
without the deformation step. This will, however, result in
noticeable artifacts (Figure 8).
Figure 7 shows the most extreme case of constraint violation (after deformation) among the examples we have processed, and an illustration of the post processing step.
∑ ∑
wtjk ||(I − Rt∗ )(v j − vk )||2 .
(16)
j,k∈t
dT
i nt <0
The area of a triangle t = (i, j, k) can be expressed as
2
t
1
2 ∑ j,k∈t w jk ||(v j − vk )|| [Hir03]. Equation 16 therefore
measures the total projected area of flipped triangles. We
use this quantity during segmentation (Section 5) to decide
if connected components should be added to a face set.
Remark An alternative approach to our deformation technique is to use a parametrization procedure. Projecting the
component along the fabrication direction and optimizing
for a bijective mapping in that plane under appropriate
boundary constraints will always lead to a valid solution. We
implemented this technique using an interior point method
suggested by Schüller et al. [SKPSH13]. However, because
Figure 8: Direct remeshing of the segmented mesh introduces noticeable artifacts (left). Constrained deformation
gives plausible results without artifacts (right).
c 2015 The Author(s)
c 2015 The Eurographics Association and John Wiley & Sons Ltd.
Computer Graphics Forum Herholz et al. / Approximating Free-form Geometry with Height Fields for Manufacturing
this method does not consider the component orthogonal to
the projection plane, geometry can get severely distorted. We
found our approach to give consistently superior results in
our application.
7. Implementation
We implemented the algorithm and the user interface in C++.
We use the software library Eigen [GJ∗ 10] for numerical
computations and CGAL [CGA] for geometric intersection
queries. To sample the face sets, faces violating (C1) and
(C2) have to be identified for each direction di . While checking for (C1) is straightforward, (C2) is more difficult due to
the global nature of the problem. To identify occluded faces,
we project all faces of the label face set onto a plane orthogonal to di and record intersecting triangles. Since directions
are processed individually the sampling can potentially be
accelerated significantly by parallelization. The energy minimization uses a modified version of code for multi-label optimization provided by Delong et al. [DOIB12].
Figure 9: The kitten model was decomposed into four pieces
using our technique and milled with a 3-axis CNC mill. After manually hollowing out the backsides, the pieces were
assembled and glued together.
7.1. Performance
The sampling of face sets Li takes the bulk of computation
time. Fortunately, this can be done in a preprocessing step.
For the model ’lion-vase’ with 200k vertices processing one
direction takes between 1.5 and 2 seconds with a mean of
1.83 seconds. For all our examples we used 600 direction,
therefore preprocessing takes about 18 minutes. The bottleneck here is the computation of 2d triangle-triangle intersections. To speed up the intersection test we use a spatial
data structure by Zomorodian et al. [ZE00] as implemented
in CGAL. Therefore, according to the analysis in [ZE00],
the sampling step scales almost linearly with respect to the
number of faces in practice.
Optimizing Eq. (5) via the α-expansion algorithm takes
about 3 minutes for the ’lion vase’ with 600 sampled directions and about 1.7 minutes for 300 directions. Using 10 iterations of ARAP deformation amounts to 5 seconds of computation time. For all examples in Figure 15 run times are
comparable to the ’lion-vase’. All timings were taken on a
3.9 GHz Quad Core machine.
8. Results
8.1. Multi-piece molds
We manufactured a number of reusable multi-piece molds
generated by our technique and used them to cast object
in resin and plaster. All results have been computed fully
automatically with the same set of parameters and without
any user interaction. Our system generated the mold pieces
by extruding seam edges and triangulating the planar back
sides.
We used a factor of λ = 103 to penalize the number of
c 2015 The Author(s)
c 2015 The Eurographics Association and John Wiley & Sons Ltd.
Computer Graphics Forum Figure 10: Segmentation into volume constrained height
field patches.
labels and a size tolerance for connected components of
flipped faces of γ = 0.05 (all meshes have been scaled to fit
into the unit cube). All results were generated with 600 adaptively sampled directions. Computation times for all examples range from 20 to 35 minutes. Mold pieces were milled
with a conventional 3-axis milling machine using a 3mm
drill bit. Results are shown in Figure 15.
8.2. Milling
Instead of using mold casting, height fields can also be
milled directly using a 3-axis milling machine. In order to assemble the parts, their backsides have to be machined manually. Since the backsides will not be visible, no special care
has to be taken in this step. Figure 9 shows an example object
assembled from milled height fields, for which the algorithm
failed to produce a valid multi-piece mold. Using our technique every 3-axis milling machine turns into an affordable
large-scale 3d printer.
8.3. Volume constraint
Using the volume constraint described in Section 4, objects
can be decomposed into height-field patches which fit into
a predefined cuboid build volume (Figure 10). This is de-
Herholz et al. / Approximating Free-form Geometry with Height Fields for Manufacturing
Figure 12: The bunny has been segmented into two parts
with limited overhang angle and printed with the Form 1
STL printer (top) and the Ultimaker 2 (bottom). No support
is needed for the surface.
sirable in situation where either the machining range or the
stock material is limited in terms of volume.
γ
Seg. Resid.
Parts
ARAP Resid.
0
0.001
0.005
0.01
0.05
0.1
0.2
2044.04
813.2
538.9
512.52
389.68
380.22
354.56
50
17
9
9
5
5
4
0
17.9906
45.127
49.0432
71.2212
107.334
147.111
Table 1: Experimental results for the ’lion vase’ model.
8.4. Influence of γ
The main parameter used to find a compromise between the
number of parts and the amount of deformation is the parameter γ. To demonstrate the impact of γ on the solution
we computed a decomposition of the model ’lion vase’ for
different parameter values and recorded the residual of the
segmentation energy (Eq. (5)) without label costs (λ = 0).
Moreover, we recorded the number of parts and the residual of the ARAP energy (Eq. (13)) to assess the amount of
non-rigid deformation due to the least squares solve. This
residual manifests itself as distortion in the mesh; triangles
might be sheared or scaled. For each γ value we ran the algorithm 20 times using random label orders and picked the
result with lowest segmentation energy (Eq. (5)). Figure 7
shows the resulting segmented and deformed models. While
values below γ = 0.05 clearly have too many segments to be
useful in practice, a value of 0.2 introduces too much distortion.
Figure 13: Depending on the segmentation, mold pieces
(colored) might block each other, even though every segment
(black) is ray-accessible (top row). If segments are not rayaccessible, mold pieces can be removable, depending on the
segmentation (center row).
8.5. Application to 3d printing
Hu et al. [HLZCO14] recently exploited the fact that decomposing a shape into parts with limited overhang angle
will save support material in 3d printing based on fused deposition modeling or stereolithography. We can modify our
segmentation algorithm slightly to save support material as
well. The goal is to segment a given shape into parts with
limited slope angle. We achieve this by adjusting the unary
energy to
(
0
if dT
li ni ≥ − cos(αhang )
(17)
Eei (li ) =
∞ otherwise.
Moreover, we have to account for the base surface of each
part since additional support material might be required for
non-planar base surfaces. We address this issue by favoring
planar seams. term. Since the segments will be fabricated
layer by layer, we do not have to account for (C2) and (C3).
The modified pairwise energy term is given by:
(
0
if k = m,
e
(18)
Ei, j (k, m) =
2
T
2
e
)
+
(d
e
)
otherwise
(dT
m i, j
k i, j
where ei, j denotes the edge connecting triangles ti and t j .
After the shape has been segmented, we triangulate the base
planes and print the segments.
9. Discussion
9.1. Global ray accessibility
As noted in Section 3.4, faces might not be ray-accessible,
meaning the face might be blocked by another face in fabrication direction. It would be straightforward to form sets
c 2015 The Author(s)
c 2015 The Eurographics Association and John Wiley & Sons Ltd.
Computer Graphics Forum Herholz et al. / Approximating Free-form Geometry with Height Fields for Manufacturing
Figure 11: Several instances of segmented meshes for γ = 0, 0.001, 0.005, 0.01, 0.05, 0.1, 0.2. By varying γ we can control the
number of faces that violate (C1). A value of zero yields no constraint violation (top, center left). Consequently many segment
are generated. Choosing γ to large yields to much deformation (bottom, very right).
Figure 14: For these more complex shapes mold pieces cannot be (dis-)assembled due to interlocking.
{L j } of faces such that each face fi ∈ L j is also rayaccessible in direction d(L j ). However, this approach seems
questionable for the following reasons:
• Valid multi-piece molds can often be generated without
requiring faces to be ray-accessible. Figure 13 (center
row) demonstrates a situation where faces are not rayaccessible w.r.t. their fabrication direction. Depending on
the segmentation, mold pieces can be disassembled (left)
or not (right). Enforcing that all faces are ray-accessible
would limit the solution space, likely resulting in more
segments for most objects.
• Even if faces are ray-accessible, it is not guaranteed that
c 2015 The Author(s)
c 2015 The Eurographics Association and John Wiley & Sons Ltd.
Computer Graphics Forum this still holds true after the deformation and remeshing
step. Incorporating accessibility constraints into the deformation is not straightforward.
• The shape of a mold piece cannot be trivially deduced
from the surface segment. For example, if the mold is
manufactured with a 3-axis mill, the shape will have a
flat base with varying height. This makes a global analysis of the assembly difficult at the stage of segmentation.
Figure 13 (bottom, right) illustrates a mold piece with a
flat base as opposed to a ’thin’ mold piece (bottom, left).
Even if using ’thin’ mold pieces, disassembly might be
impossible, depending on the segmentation (top, left).
9.2. Limitations
In its current form, our algorithm is limited to fairly simple
shapes, possibly with small scale details. For more complex
geometry our system might fail to produce mold pieces that
can be assembled or disassembled (Figures 13, 14) or might
generate too many pieces to be useful in practice. While we
do believe that some shapes are not suitable for heightfieldbased fabrication, we hope to improve on our result in terms
of a better deformation algorithm. Simplifying the topology,
for example, might help in finding a manufacturable approximation of the input shape.
Herholz et al. / Approximating Free-form Geometry with Height Fields for Manufacturing
9.3. Conclusion
We proposed a method to fabricate a freeform geometry with classical manufacturing techniques such as 3-axis
milling, casting, and stamping. These techniques are limited to shapes that are height fields. To nonetheless fabricate
more general objects we segment them into multiple components that are manufactured separately and then assembled.
Strictly enforcing the height field constraint for each component leads to a prohibitively large number of pieces. During
segmentation we hence allow small “islands” that violate the
constraint in each part. After segmentation these violations
are fixed using a deformation step. We employed a variation
of the as-rigid-as-possible approach to obtain these deformations. Our technique can also be employed to improve
3d printing by reducing the need for support material or enabling to place it in inconspicuous places.
We fabricated molds and objects for various 3d models.
The results demonstrate the practicality of our approach. In
particular, we typically obtain only a small number of pieces
with a deformation that is not noticeable without comparison.
Acknowledgments
The authors would like to thank Jacob Seibert for his help
with mold manufacturing and casting. We are also grateful
for the constructive reviewer comments that helped us improve the paper. The meshes are courtesy of AIM@SHAPE,
Thingiverse and Stanford Computer Graphics Lab. This
work has been supported by the ERC through grant ERC2010-StG 259550 ("XSHAPE").
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c 2015 The Author(s)
c 2015 The Eurographics Association and John Wiley & Sons Ltd.
Computer Graphics Forum Herholz et al. / Approximating Free-form Geometry with Height Fields for Manufacturing
Figure 15: Several multi-piece molds were generated using our method. Based on the segmented input meshes (first column),
the meshes were deformed (second column, scale distortion color coded) and mold pieces were fabricated using CNC milling
(last column). These molds where assembled and filled with liquid resin to produce physical replicas (right center column).
c 2015 The Author(s)
c 2015 The Eurographics Association and John Wiley & Sons Ltd.
Computer Graphics Forum